= Stationary law of a linear flow network
{title2=$\pi(n)\propto\binom{\sum_rn_r}{n_0}\prod_r\rho_r^{n_r}$}
For the <proportionally fair allocation on a linear flow network>, independent Poisson flow arrivals at rates $\nu_r$ and exponential document sizes of rates $\mu_r$ give a <continuous-time Markov chain> with departures $\mu_rn_rx_r(n)$. Put $\rho_r=\nu_r/\mu_r$. Under $\rho_0+\rho_i<1$ for each local route, the normalizing constant is $Z=(1-\rho_0)^{I-1}/\prod_i(1-\rho_0-\rho_i)$. The <binomial coefficient> weight has adjacent-state ratios equal to the aggregate service rates, proving <detailed balance>. For more general network topologies, <proportional fairness> need not give this reversible product form.
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